Big Idea:
Students can make sense of the types of values a variable has through set notation.

I find that it helps students from being overwhelmed with set builder notation if we break up the introduction of the format into discrete pieces. As my Set Builder Notation presentation illustrates, I start with the brackets:

Set builder notation always starts with two squiggly wiggly brackets, like this:

{ }

Then I introduce the variable.

Set builder notation that starts by introducing the variable letter. Basically you are telling your reader what letter is being used as a variable. This can be anything you want, but it typically is introduced first, like this:

{ x }

Next, I explain why the link appears after the letter.

We write a short vertical line to separate the variable from the rest of the statement:

{ x | }

If you are reading this to yourself, you could already say something like, ‘given the variable x, such that the following is true.’ That vertical line is read as such that and tells the reader that you are about to define the variable.

Next, I introduce the interval notation.

The goal in set builder notation is to define the variable in some way. Remember that a variable could be any value or any group of values. So we need to tell our reader what domain this variable lives in. Is it positive? Negative? A fraction? A big number? A small number? What is it?

I write the interval notation and ask them to write down what they think it means:

{ x | -3 ≤ x ≤ 3 }

Many students aren’t sure how to read this notation. They might write something like “-3 is less than or equal to x and 3” or “3 is greater than or equal to x and -3 is less than or equal to x.” Although these aren’t wrong, I thin kit is important to help them find a way to read the expression to focus on the value of the variable.

What does this notation tell us about the variable x? What values does it live between?

Once we shift the focus to the variable, students get a sense of the domain for the variable x. They start to understand that the best way to read this notation is “x is bigger than or equal to -3 and less than or equal to 3.” More concisely, students say “x is between -3 and +3, inclusive.”

Since many of them have used inclusive and exclusive notation before (on a number line) I do a quick review.

How can we graph this on a number line?” We set up a number line and draw two closed circles over -3 and +3 and connect them with a line. I also show them bracket notation, where the square brackets “] or [“ mean exclusive and the curved brackets “) or (“ mean inclusive. In this case we would write [-3,+3].

Then we consider alternative situations.

What if we had -3<x<3?

Here we have two open circles and two curved brackets.

What if we had -3 < x ≤ 3?

Here we have an open circle followed by a closed circle, or (-3,+3]

What if we have -3 ≤ x < 3?

Here we have a closed circle followed by an open circle, or [-3,+3)

At this point, if students are confused as to when to use the open or closed circle, I will change the language they use to describe these shapes. Using the word “closed” implies exclusive and is thus confusing. Instead I might focus on calling one circle “filled in” and the other “empty.” This language matches the implication that a “filled in” circle includes the value and an “empty” circle excludes the value. From there it is possible to generate all types of analogies, “when you see a filled in circle, think of inclusion, like inviting everyone over to fill the space. When you see an empty circle, think of exclusive and loneliness.” These types of creative analogies have really helped many students reason through the meaning of the dots long after we have given these lessons.

I find that I have less success with the bracket notation, but it seems to help somewhat to think that the square bracket would hold more than the curved bracket and thus include the value it is next to.

This discussion on interval notation brings up an important point for the students: the interval chosen (the domain) really changes the definition of the variable, especially on the end points of the interval.

Getting back to set notation, we finish with writing out the group in which the variable lives.

If x is between -3 and +3, there are an infinite number of possible values to choose from if we are using rational or real numbers. But what if our variable could only be an integer? What if it could only be a natural number? How many values could we then use for x?

Here we discuss the importance of knowing what type of number x is. Students understand that natural or counting numbers would be the most restrictive, followed by integers and so on. I finish the introduction by showing how to include this in set builder notation:

Resources (1)

Resources

Students work through the various set builder problems while I circulate and gather interesting algorithms and common misconceptions. I ask that they not only answer the set builder questions by circling the correct choice, but that they also list out all possible values for x in each scenario. I find this to be very important for the summary

In this problem they tell students that our set includes the number {11,12} and then asks for the correct notation to match. I have students present all possible values for x, based on the notation written in each choice. Since the numbers are so close, it really pushes students thinking on the importance of inclusivity and exclusivity. I love when they realize that the second choice is written in a way to only include one value! This surprises them and lets me know that they are really understanding the topic.